Vertex algebras with big center and a Kazhdan-Lusztig Correspondence

Certain deformable families of vertex algebras acquire at a limit of the deformation parameter a large center, similar to affine Lie algebras at critical level. Then the vertex algebra and its representation category become a bundle over the variety defined by this large center. The zero-fibre becomes a vertex algebra, the other fibres become twisted modules over this vertex algebra. We explore these ideas and a conjectural correspondence in a class of vertex algebras $A^{(p)}[\mathfrak{g},\kappa]$ associated to a choice of a finite-dimensional semisimple Lie algebra and an integer $(\mathfrak{g},p)$ and a level $\kappa$, and some related algebras. These algebras were introduced under the name quantum geometric Langlands kernels and have an interpretation in 4-dimensional quantum field theory. In the limit $\kappa\to \infty$, they acquire a large central subalgebra identified with the ring of functions on the space of $\mathfrak{g}$-connections. The zero fibre is expected to be the Feigin-Tipunin algebra $W_p(\mathfrak{g})$, whose category of representations is expected to be equivalent to the small quantum group by the logarithmic Kazhdan Lusztig conjecture. Our rigorous results focus on the cases $(\mathfrak{g},1)$ and $(\mathfrak{sl}_2,2)$. In the first part of the paper, which might be of independent interest, we discuss the twisted modules (including twists by unipotent elements) of the affine Lie algebra and of the triplet algebra $W_2(\mathfrak{sl}_2)$, and introduce a method of twisted free field realization to obtain twisted modules for arbitrary $(\mathfrak{g},p)$. We also match our findings to the respective quantum group with big center. In the second part of the paper we match the results to the vertex algebra bundle we compute from the limit of $A^{(p)}[\mathfrak{g},\kappa]$, which in our cases have realizations as GKO coset resp. N=4 superconformal algebras.